{"658513":{"#nid":"658513","#data":{"type":"event","title":"PhD Defense by Xiaofan Yuan","body":[{"value":"\u003Cp\u003EXiaofan Yuan\u003Cbr \/\u003E\r\n(Advisor: \u0026nbsp;Prof. \u0026nbsp;Xingxing Yu) will defend a doctoral thesis entitled,\u003Cbr \/\u003E\r\nMatching Problems in Hypergraphs\u003Cbr \/\u003E\r\nOn\u003Cbr \/\u003E\r\nThursday, June 9 at 10:00 a.m.\u003Cbr \/\u003E\r\nSkiles 006\u003Cbr \/\u003E\r\nhttps:\/\/gatech.zoom.us\/j\/91659544858?pwd=SWZtVG15dGFiWEFXSHR1U0JNbVVBZz09\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cbr \/\u003E\r\nAbstract\u003Cbr \/\u003E\r\nKu\u0026uml;hn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph\u0026nbsp;\u003Cbr \/\u003E\r\nwith n vertices, where n \u0026isin; 3Z and large, and \u0026delta;\u2081(H) \u0026gt; \u0026nbsp;\u207f\u0026minus;\u0026sup1; \u0026nbsp;\u0026minus; 2n\/3 \u0026nbsp;, then H contains a perfect\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cbr \/\u003E\r\n\u0026nbsp;\u0026nbsp; \u0026nbsp;\u0026nbsp;\u0026nbsp; \u0026nbsp;\u003Cbr \/\u003E\r\nWe show that for n \u0026nbsp; \u0026nbsp; 3Z \u0026nbsp;sufficiently large, \u0026nbsp;if F , . . . , F \u0026nbsp; \u0026nbsp; \u0026nbsp;are 3-uniform hypergraphs\u0026nbsp;\u003Cbr \/\u003E\r\nwith a common vertex set and \u0026delta;\u2081(Fi) \u0026gt; (n\u0026minus;1)\u0026minus;(2n\/3) for i \u0026isin; [n\/3], then {F\u2081, . . . , Fn\/3} admits a\u0026nbsp;\u003Cbr \/\u003E\r\nrainbow\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cbr \/\u003E\r\n\u0026nbsp;\u0026nbsp; \u0026nbsp;\u003Cbr \/\u003E\r\nmatching, i.e., a matching consisting of one edge from each Fi. \u0026nbsp;This is done by converting the\u003Cbr \/\u003E\r\nrainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.\u0026nbsp;\u003Cbr \/\u003E\r\nWe also prove that, for any integers k, l with k \u0026ge; 3 and k\/2 \u0026lt; l \u0026le; k \u0026minus; 1, there exists a positive\u0026nbsp;\u003Cbr \/\u003E\r\nreal \u0026micro; such that, for all sufficiently large integers m, n satisfying\u003Cbr \/\u003E\r\nn\u003Cbr \/\u003E\r\nk\u003Cbr \/\u003E\r\n\u0026minus; \u0026micro;n \u0026le; m\u003Cbr \/\u003E\r\n\u0026le;\u003Cbr \/\u003E\r\nn\u003Cbr \/\u003E\r\nk\u003Cbr \/\u003E\r\n\u0026minus;\u003Cbr \/\u003E\r\n1\u003Cbr \/\u003E\r\nl\u003Cbr \/\u003E\r\n\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; l\u003Cbr \/\u003E\r\n2l \u0026minus; k\u003Cbr \/\u003E\r\n,\u003Cbr \/\u003E\r\nif H \u0026nbsp;is a k-uniform hypergraph on n \u0026nbsp;vertices and \u0026delta;l(H) \u0026nbsp;\u0026gt;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E(n\u0026minus;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cbr \/\u003E\r\n) \u0026minus;\u003C\/p\u003E\r\n\r\n\u003Cp\u003E) \u0026nbsp; hen H \u0026nbsp;has a\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u003Cbr \/\u003E\r\n. This improves upon an earlier result of H`an, Person, and Schacht for the range k\/2 \u0026nbsp;\u0026lt; l \u0026le; k \u0026minus; 1.\u0026nbsp;\u003Cbr \/\u003E\r\n\u0026nbsp;In many cases, our result gives tight bound on \u0026delta;l(H) \u0026nbsp;for near perfect matchings (e.g., \u0026nbsp;when l \u0026nbsp;\u0026ge;\u0026nbsp;\u003Cbr \/\u003E\r\n2k\/3, \u0026nbsp;n \u0026nbsp;\u0026equiv; r \u0026nbsp;(mod k), \u0026nbsp;0 \u0026nbsp;\u0026le; r \u0026nbsp;\u0026lt; \u0026nbsp;k, \u0026nbsp;and r + l \u0026nbsp;\u0026ge; k, \u0026nbsp;we can take m = \u0026nbsp;n\/k \u0026nbsp;\u0026minus; 2).\u003Cbr \/\u003E\r\nCommittee\u003Cbr \/\u003E\r\n\u0026bull; \u0026nbsp;Prof. Anton Bernshteyn - School of Mathematics, Georgia Institute of Technology\u003Cbr \/\u003E\r\n\u0026bull; \u0026nbsp;Prof. Hao Huang - Department of Mathematics, National University of Singapore (reader)\u003Cbr \/\u003E\r\n\u0026bull; \u0026nbsp;Prof. Santosh Vempala - College of Computing, Georgia Institute of Technology\u003Cbr \/\u003E\r\n\u0026bull; \u0026nbsp;Prof. Josephine Yu - School of Mathematics, Georgia Institute of Technology\u003Cbr \/\u003E\r\n\u0026bull; \u0026nbsp;Prof. Xingxing Yu - School of Mathematics, Georgia Institute of Technology (advisor)\u003Cbr \/\u003E\r\n\u0026nbsp;\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Matching Problems in Hypergraphs"}],"uid":"27707","created_gmt":"2022-05-25 19:10:37","changed_gmt":"2022-05-25 19:10:37","author":"Tatianna Richardson","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2022-06-09T11:00:00-04:00","event_time_end":"2022-06-09T13:00:00-04:00","event_time_end_last":"2022-06-09T13:00:00-04:00","gmt_time_start":"2022-06-09 15:00:00","gmt_time_end":"2022-06-09 17:00:00","gmt_time_end_last":"2022-06-09 17:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}